Concave and positive summing norms for operators on \(L^p\) (Q5927597)

For an operator \(T_1\) from \(L^p(\lambda _1)\) into a Banach space \(X\) where the conjugate of \(p\) is not an even integer, it is shown that the image of the unit ball determines if the operator is concave or positive summing. Specifically, for an operator \(T_1\) as above and another operator \(T_2\) from \(L^p(\lambda _2)\) into \(X\) , if the closure of the image of the unit balls is equal, then \(T_1\) satisfies the condition of \((r,s)\)-concave or positive \((r,s)\)-summing if and only if \(T_2\) satisfies the condition. Further, if the closure of the image of the unit balls are equal, then indeed the concave norms coincide and conditions on the summing norms are analyzed. Similar considerations are used to show that the image of the unit ball determines representability of the operator with respect to a function in \(L^{p'}(\lambda ,X)\) where \(p'\) is the conjugate.